18.099b Problem Set 3b Due: Thursday, Match 3rd (in class or before).

Let X be a metric space. Recall that a neighbourhood is a subset of X of the form

N_{r}(p) := {x ∈ X : d(p, x) < r}

where r is a positive real number and p is a point in X. The number r is called the radius of N_{r}(p).

Suppose every infinite subset of X has a limit point. Prove that for every positive real number r > 0 there are finitely many neighbourhoods of radius r that cover X.

Hint: Assume, for a contradiction, that the conclusion fails for some r > 0. Construct an infinite set {p_{1}, p_{2}, … } in X such that for any i ≠ j d(p_{i}, p_{j}) ≥ r. Use this to contradict our assumption on X.